Some function spaces on symmetric spaces related to convolution operators (Q760871)

Let \(\mu\) be a positive measure on a semi-simple Lie group G equiped with a Haar measure. The authors obtain a necessary and sufficient condition on \(\mu\) such that it defines a bounded operator on \(L^ p(G)\) by convolutions under the assumption that \(\mu\) is invariant with respect to the action of the maximal compact subgroup K. By considering \(\mu\) as a measure on a symmetric space \(X=G/K\), the authors obtain a theorem similar to Kunz-Stein which states that if \(1\leq p<2\), every \(L^ p(G)\) function defines by convolution a bounded operator on \(L^ 2(G)\).